Question

The numbers of $ 3 \times 3 $ matrices A whose entries are either $ 0 $ or $ 1 $ and for which the system A $ \left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right] $ has exactly two distinct solutions?
A. $ 0 $
B. $ {2^9} - 1 $
C. $ 168 $
D. $ 2 $

Answer 1

Hint: Here first of all we will suppose any third order matrix and then will place it in the given conditions and then will simplify the equations finding the product first and then will find out the values accordingly.

Complete step by step solution:
Let us assume any $ 3 \times 3 $ matrix be
 $ A = \left[ {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{b_1}}&{{b_2}}&{{b_3}} \\
  {{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right] $
Where these elements can be $ 0 $ or $ 1 $
We are also given that A $ \left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right] $
Place the values for matrix A
 $ \left[ {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{b_1}}&{{b_2}}&{{b_3}} \\
  {{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right] $
Apply the law for the product of two matrices in the above expression.
 $ \left[ {\begin{array}{*{20}{c}}
  {{a_1}x + {a_2}y + {a_3}z} \\
  {{b_1}x + {b_2}y + {b_3}z} \\
  {{c_1}x + {c_2}y + {c_3}z}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right] $
Comparing both the sides of the matrix with respective rows on both the sides of the equation.
 $
  {a_1}x + {a_2}y + {a_3}z = 1 \\
  {b_1}x + {b_2}y + {b_3}z = 0 \\
  {c_1}x + {c_2}y + {c_3}z = 0 \;
  $
The above equation implies that-
 $ {b_1} = {b_2} = {b_3} = {c_1} = {c_2} = {c_3} = 0 $
And $ {a_1}x + {a_2}y + {a_3}z = 1 $
This set of equations cannot have exactly two distinct solutions and that is not possible.
Hence, there is no such A matrix satisfying the given conditions.
Therefore, from the given multiple choices, option A is the correct answer.
So, the correct answer is “Option A”.

Note: Be careful while finding the product of two matrices and also know that the product of the matrix only exists when the total number of rows of the first matrix is equal to the number of columns in the second matrix.
Always remember that the product of two matrices is only possible when the inner dimensions are equal, which means that the number of columns of the first matrix is equal to the number of the rows of the second matrix. If the product of the matrix does not exist then the inverse of the product of the two matrices also does not exist. Also, the determinant of the matrix should be not equal to zero to get its inverse.